Multiplication and Inversion Formulas for Higher-Order Lerch Zeta Functions with Polynomial Coefficients

1. Introduction and Preliminaries

We begin by recalling the notation and definitions used throughout the paper. For additional background and related results, we refer to [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22].

1.1. Higher Hurwitz-Lerch Zeta Functions

Let $\lambda$ and x be complex numbers such that $\left|\lambda \right|\le 1$ and $\Re \left(x\right)>0$. The Lerch transcendent function is given by the series

$$\begin{array}{c}\hfill \Phi (\lambda ,s,x)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{\lambda }^k}{{(x+k)}^s}},\end{array}$$

(1)

see ([11] §1.11, p. 27), ([3] §25.14), [12] and [14,15].

Let N be a positive integer, the higher-order Hurwitz-Lerch zeta function of order N (or N-tuple) is defined by

$$\begin{array}{c}\hfill {\zeta }_N(s,x,\lambda )=\sum _{k_1,\dots ,k_N\ge 0}{\displaystyle \frac{{\lambda }^{k_1+\cdots +k_N}}{{(x+k_1+\cdots +k_N)}^s}},\end{array}$$

(2)

where $x\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$, $s\in \mathbb{C}$ when $\left|\lambda \right|<1;\Re \left(s\right)>n$ when $\left|\lambda \right|=1$ see also ([21] p. 201).

By use of the identity ${\displaystyle {(x+k)}^{-s}=\frac{1}{\Gamma \left(s\right)}{\int }_0^{\infty }{t}^{s-1}{e}^{-(x+k)t}dt,}$ we obtain the following integral representation

$$\begin{array}{c}\hfill {\displaystyle {\zeta }_N(s,x,\lambda )=\frac{1}{\Gamma \left(s\right)}{\int }_0^{\infty }\frac{t^{s-1}{e}^{-xt}}{{(1-\lambda e^{-t})}^N}dt,}\end{array}$$

(3)

where $\Re \left(s\right)>N$ for $\lambda =1$, and $\Re \left(s\right)>0$ for $\lambda \ne 1$ see [4,6] also [21].

Let $k,k_1,\dots ,k_N$ be non-negative integers, the number of solutions in $k_1+\cdots +k_N=k$ is equal to $\left({\displaystyle \genfrac{}{}{0pt}{}{k+N-1}{N-1}}\right)$. Hence, the higher Hurwitz zeta function in (2) can be represented by the interesting serie

$$\begin{array}{c}\hfill {\zeta }_N(s,x,\lambda )={\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{k+N-1}{N-1}}\right){\displaystyle \frac{{\lambda }^k}{{(x+k)}^s}}\end{array}$$

(4)

see also [21]. Note that $\Phi (\lambda ,s,x)={\zeta }_1(s,x,\lambda )$. In the remainder of the paper, we denote

$$\zeta (s,x,\lambda ):={\zeta }_1(s,x,\lambda )=\Phi (\lambda ,s,x)$$

see also [21].

1.2. Higher-Order Apostol-Bernoulli and Apostol-Euler Polynomials and Polynomials Coefficients

The Apostol-Euler type polynomials $E_n^{\left(N\right)}(x;\lambda )$ of order N and parameter $\lambda \ne -1$, are defined by the generating functions

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{2}{1+\lambda e^t}}\right)}^N{e}^{xt}={\displaystyle \sum _{n=0}^{\infty }}{E}_n^{\left(N\right)}(x;\lambda ){\displaystyle \frac{t^n}{n!}}\left(\right|t|<|log(-\lambda )\left|\right)\end{array}$$

see also [21]. The Apostol-Bernoulli type polynomials $B_n^{\left(N\right)}(x;\lambda )$ of order N and parameter $\lambda$, are defined by the generating functions

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{t}{\lambda e^t-1}}\right)}^N{e}^{xt}={\displaystyle \sum _{n=0}^{\infty }}{B}_n^{\left(N\right)}(x;\lambda ){\displaystyle \frac{t^n}{n!}},\end{array}$$

where $\left|t\right|<2\pi$ for $\lambda =1$ and $\left|t\right|<|log\lambda |$ for $\lambda \ne 1$. For more details see, for instance, [2,15].

The function $s\to {\zeta }_N(s;x,\lambda )$ has analytic continuation to whole complex numbers, except eventual poles at $s=1,\cdots ,N.$ At $s=-n<0$ negative integers, we have the relations

$$E_n^{\left(N\right)}(x;\lambda )=2^N{\zeta }_N(-n,x,-\lambda )$$

(5)

and

$$B_{n+N}^{\left(N\right)}(x;\lambda )={(-1)}^N<n+1{>}_N{\zeta }_N(-n,x,\lambda ),$$

(6)

where $<x{>}_n$ is the rising factorial given by

$$\begin{array}{c}\hfill <x{>}_0=1,<x{>}_N=x(x+1)\cdots (x+N-1),N\ge 1\end{array}$$

see also [13,21]. These polynomials satisfy the following symmetries and binomial relations

$${\lambda }^N{B}_n^{\left(N\right)}(N-x;\lambda )={(-1)}^n{B}_n^{\left(N\right)}(x;{\lambda }^{-1}),{\lambda }^N{E}_n^{\left(N\right)}(N-x;\lambda )={(-1)}^n{E}_n^{\left(N\right)}(x;{\lambda }^{-1})$$

and

$$B_n^{\left(N\right)}(x;\lambda )={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{n}}\right)B_k^{\left(N\right)}\left(\lambda \right)x^{n-k},E_n^{\left(N\right)}(x;\lambda )={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{n}}\right)E_n^{\left(N\right)}\left(\lambda \right)x^{n-k},$$

where $B_k^{\left(N\right)}\left(\lambda \right)=B_k^{\left(N\right)}(0;\lambda )$ and $E_k^{\left(N\right)}\left(\lambda \right)=E_k^{\left(N\right)}(0;\lambda )$ are the Apostol-Bernoulli and Apostol-Euler numbers respectively, see also [5,21].

To state the results of this paper, we need the following multinomial theorem and the theorem on polynomial coefficients. For any positive integer m and any non-negative integer n, the multinomial theorem describes how a sum with m terms expands when raised to the power:

$${\displaystyle {(x_1+x_2+\cdots +x_m)}^n=\sum _{\begin{array}{c}{k}_1+\cdots +k_m=n\\ k_1,\cdots ,k_m\ge 0\end{array}}\left(\genfrac{}{}{0pt}{}{n}{k_1,\dots ,k_m}\right)x_1^{k_1}\cdots x_m^{k_m},}$$

where $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k_1,\dots ,k_m}}\right)={\displaystyle \frac{n!}{k_1!\cdots k_m!}}$ is called the multinomial coefficient. The sum is taken over all m-tuples of non-negative integer $k_1,\cdots ,k_m$ such that $k_1+\cdots +k_m=n$.

For integers $N,m,k$ with $N\ge 1$, $m\ge 2$, and $0\le k\le N(m-1)$, Comtet [10, Problem 16, pp.77-78] introduced the polynomials coefficients $\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right)$ by

$$\begin{array}{c}\hfill {(1+t+\cdots +t^{m-1})}^N={\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right)t^k,\end{array}$$

(7)

equivalently by the corresponding multinomial expansion

$${\displaystyle \left(\genfrac{}{}{0pt}{}{N,m}{k}\right)=\sum _{k_1,+2k_2+\cdots +(m-1)k_{m-1}=k}\left(\genfrac{}{}{0pt}{}{n}{k_1,k_2,\dots ,k_{m-1}}\right).}$$

From a combinatorial point of view $\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right)$ is exactly the number of k-combinations of $\{0,1,\dots ,N\}$ with multiplicities less than m.

Note that

$$\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right)=0\mathrm{for}k<0\mathrm{or}k>N(m-1).$$

For the special cases $m=2$ and $m=\infty$, we obtain the binomial coefficients

$$\left({\displaystyle \genfrac{}{}{0pt}{}{N,2}{k}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{N}{k}}\right),\left({\displaystyle \genfrac{}{}{0pt}{}{N,\infty }{k}}\right)={(-1)}^k〈\begin{array}{c}N\\ k\end{array}〉,$$

where $〈\begin{array}{c}N\\ k\end{array}〉={\displaystyle \frac{<N{>}_k}{k!}}$. Moreover, we have

$${\displaystyle \left(\genfrac{}{}{0pt}{}{N,m}{k}\right)=\sum _{mi+j=k}\left(\genfrac{}{}{0pt}{}{N}{i}\right)〈\begin{array}{c}N\\ j\end{array}〉.}$$

2. New Results on Higher Hurwitz-Lerch Zetas

In this section, we state the main results and precise some corollaries and remarks. First, we describe ${\zeta }_N(s+N,x,\lambda )$ in terms of Bernoulli polynomials $B_k^{\left(N\right)}\left(x\right)$ of order N and Hurwitz-Lerch zeta function $\Phi (\lambda ,s,x).$

Theorem 1

(Reduction theorem). Under the conditions of its definition, ${\zeta }_N$ satisfies

$$\begin{array}{c}\hfill (N-1)!{\zeta }_N(s+N,x,\lambda )={\displaystyle \sum _{j=1}^N}{(-1)}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{N-1}{j-1}}\right)B_{j-1}^{\left(N\right)}\left(x\right)\Phi (\lambda ,s+j,x).\end{array}$$

(8)

Proof. 

From Nörlund ([16] Formula $\left(86\right)$, p.147) we have, for any integer $0\le j\le N$,

$$\begin{array}{c}\hfill B_j^{(N+1)}\left(x\right)={\displaystyle \frac{j!}{N!}}{D}^{N-j}{(x-1)}_N,\end{array}$$

thus, for any parameters x, a, we have

$$\begin{array}{c}\hfill {(a-1)}_N={\displaystyle \sum _{j=0}^N}{\displaystyle \frac{{(a-x)}^{N-j}}{(N-j)!}}{D}^{N-j}{(x-1)}_N={\displaystyle \sum _{j=0}^N}\left({\displaystyle \genfrac{}{}{0pt}{}{N}{j}}\right)B_j^{(N+1)}\left(x\right),{(a-x)}^{N-j},\end{array}$$

we after choose $a=-k$, where k is an integer and $k\le N$ and hold

$$\begin{array}{c}\hfill {(-k-1)}_N={(-1)}^N(k+1)(k+2)\dots (k+N)={\displaystyle \sum _{j=0}^N}\left({\displaystyle \genfrac{}{}{0pt}{}{N}{j}}\right){(-1)}^{N-j}{B}_j^{(N+1)}\left(x\right){(x+k)}^{N-j}\end{array}$$

that is

$$\begin{array}{c}\hfill N!\left({\displaystyle \genfrac{}{}{0pt}{}{N+k}{N}}\right)={\displaystyle \sum _{j=0}^N}\left({\displaystyle \genfrac{}{}{0pt}{}{N}{j}}\right){(-1)}^j{B}_j^{(N+1)}\left(x\right){(x+k)}^{N-j}\end{array}$$

and therefore, we have

$$\begin{array}{ccc}\hfill N!{\zeta }_{N+1}(s+N,x,\lambda )& =& {\displaystyle \sum _{k=0}^{\infty }}N!\left({\displaystyle \genfrac{}{}{0pt}{}{N+k}{N}}\right){\displaystyle \frac{{\lambda }^k}{{(x+k)}^{s+N}}}\hfill \\ & =& {\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \sum _{j=0}^N}\left({\displaystyle \genfrac{}{}{0pt}{}{N}{j}}\right){(-1)}^j{B}_j^{(N+1)}\left(x\right){(x+k)}^{N-j}\right){\displaystyle \frac{{\lambda }^k}{{(x+k)}^{s+N}}}\hfill \\ & =& {\displaystyle \sum _{j=0}^N}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{N}{j}}\right)B_j^{(N+1)}\left(x\right){\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{\lambda }^k}{{(x+k)}^{s+j}}}\hfill \\ & =& {\displaystyle \sum _{j=0}^N}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{N}{j}}\right)B_j^{(N+1)}\left(x\right)\Phi (\lambda ,s+j,x).\hfill \end{array}$$

Hence, we get

$$\begin{array}{c}\hfill (N-1)!{\zeta }_N(s+N,x,\lambda )={\displaystyle \sum _{j=1}^N}{(-1)}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{N-1}{j-1}}\right)B_{j-1}^{\left(N\right)}\left(x\right)\Phi (\lambda ,s+j,x).\end{array}$$

□

We note that the coefficients in the identity (8) do not depend on $\lambda$ and s, and are polynomials in x. Next, we give a few explicit expressions for ${\zeta }_N(s,x,\lambda )$. For $N=2,3,4$ we have

$$\begin{array}{ccc}\hfill {\zeta }_2(s,x,\lambda )& =& (1-x)\zeta (s,x,\lambda )+\zeta (s-1,x,\lambda ),\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill {\zeta }_3(s,x,\lambda )& =& {\displaystyle \frac{1}{2}}(x^2-3x+2)\zeta (s,x,\lambda )\hfill \\ & & +\left({\displaystyle \frac{3}{2}}-x\right)\zeta (s-1,x,\lambda )+{\displaystyle \frac{1}{2}}\zeta (s-1,x,\lambda ),\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill {\zeta }_4(s,x,\lambda )& =& {\displaystyle \frac{1}{6}}(-x^3+6x^2-11x+6)\zeta (s,x\lambda )\hfill \\ & & +(3x^2-12x+11)\zeta (s-1,x,\lambda )-(3x-6)\zeta (s-2,x,\lambda )+\zeta (s-3,x,\lambda ).\hfill \end{array}$$

(11)

Theorem 2

(Multiplication formula). Let x be a positive real. For any positive integers $m,N$, we have

$$\begin{array}{c}\hfill {\zeta }_N(s,mx,\lambda )=m^{-s}{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{\zeta }_N(s,x+{\displaystyle \frac{k}{m}},{\lambda }^m),\end{array}$$

(12)

for $\Re \left(s\right)>N$ or $\Re \left(s\right)>0$ according to ${\lambda }^m=1$ or ${\lambda }^m\ne 1.$

First proof.

We begin from the integral representation. Substituting $x\mapsto mx$ into (3), we obtain

$${\zeta }_N(s,mx,\lambda )={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }{t}^{s-1}{e}^{-mxt}{(1-\lambda e^{-t})}^{-N}dt.$$

Let $u=mt\Rightarrow t={\displaystyle \frac{u}{m}},dt={\displaystyle \frac{du}{m}}$:

$${\zeta }_N(s,mx,\lambda )={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }{\left({\displaystyle \frac{u}{m}}\right)}^{s-1}{e}^{-xu}{(1-\lambda e^{-u/m})}^{-N}{\displaystyle \frac{du}{m}}.$$

Then,

$${\zeta }_N(s,mx,\lambda )=m^{-s}{\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }{u}^{s-1}{e}^{-xu}{(1-\lambda e^{-u/m})}^{-N}du.$$

Now using generating function of polynomial coefficients:

$${(1-\lambda e^{-u/m})}^{-N}={\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{e}^{-ku/m}{(1-{\lambda }^m{e}^{-u})}^{-N}.$$

Substituting into the integral:

$${\zeta }_N(s,mx,\lambda )=m^{-s}{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }{u}^{s-1}{e}^{-(x+{\displaystyle \frac{k}{m}})u}{(1-{\lambda }^m{e}^{-u})}^{-N}du,$$

where

$${\zeta }_N\left(s,x+{\displaystyle \frac{k}{m}},{\lambda }^m\right)={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }{u}^{s-1}{e}^{-(x+{\displaystyle \frac{k}{m}})u}{(1-{\lambda }^m{e}^{-u})}^{-N}du.$$

Thus, we arrive at the desired result.

$${\zeta }_N(s,mx,\lambda )=m^{-s}{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{\zeta }_N\left(s,x+{\displaystyle \frac{k}{m}},{\lambda }^m\right).$$

□

Second proof.

Let m be a positive integer. We have

$$\begin{array}{ccc}\hfill {\zeta }_N(s,mx,\lambda )& =& \sum _{k_1,..,k_N\ge 0}{\displaystyle \frac{{\lambda }^{k_1+\dots +k_N}}{{(mx+k_1+..+k_N)}^s}}\hfill \\ & =& m^{-s}{\displaystyle \sum _{i_1,..,i_N=0}^{m-1}}{\lambda }^{i_1+...+i_N}{\zeta }_N(s,x+{\displaystyle \frac{i_1+\dots +i_N}{m}},{\lambda }^m),\hfill \end{array}$$

since

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right)=\mathrm{card}\{(i_1,..,i_N)\in {\mathbb{N}}^N:0\le i_1,\cdots ,i_N\le m-1,i_1+\cdots +i_N=k\}=p_{N,m}\left(k\right),\end{array}$$

where $p_{N,m}\left(k\right)$ is called the restricted partition function of k.

Hence, we obtain the desired formula. As $\left({\displaystyle \genfrac{}{}{0pt}{}{1,m}{k}}\right)=1$ for $k=0,1,\cdots ,m-1$, we see that for $N=1$, we recover the well-known multiplication formula for the Lerch zeta function $\Phi$, proved in ([15] (3.8)) and ([9] p.339). □

Third proof.

From the identity, see in [21],

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{(1-t^m)}^N}}{\left({\displaystyle \frac{1-t^m}{1-t}}\right)}^N={\displaystyle \frac{1}{{(1-t)}^N}}\end{array}$$

we get

$$\begin{array}{c}\hfill {\displaystyle \sum _{k,l=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{N-1+l}{N-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right)t^{ml+k}={\displaystyle \sum _{n=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{N-1+n}{N-1}}\right)t^n\end{array}$$

and then we have

Lemma 1.

Let $m,n$ and N non-negative integers. We have

$$\begin{array}{c}\hfill {\displaystyle \sum _{ml+k=n}\left(\genfrac{}{}{0pt}{}{N+l}{N}\right)\left(\genfrac{}{}{0pt}{}{N+1,m}{k}\right)=\left(\genfrac{}{}{0pt}{}{N+n}{N}\right).}\end{array}$$

Now, we apply Lemma 1 as follows. Put $D_{\lambda }:={\displaystyle \frac{\partial }{\partial \lambda }}$ , by use of series representations of zetas and multiplication formula for $\Phi$, we have

$$\begin{array}{ccc}\hfill {\zeta }_N(s,mx,\lambda )& =& {\displaystyle \frac{1}{(N-1)!}}{D}_{\lambda }^{N-1}\left({\lambda }^{N-1}\Phi (s,mx,\lambda )\right)\hfill \\ & =& {\displaystyle \sum _{j=0}^{m-1}}{\displaystyle \frac{1}{(N-1)!}}{D}_{\lambda }^{N-1}\left({\lambda }^j\Phi (s,x+j/m,{\lambda }^m)\right).\hfill \end{array}$$

This gives

$$\begin{array}{ccc}\hfill {\zeta }_N(s,mx,\lambda )& =& m^{-s}{\displaystyle \sum _{j=0}^{m-1}}{\displaystyle \sum _{n=0}^{\infty }}{(x+j/m+n)}^{-s}{\displaystyle \frac{1}{(N-1)!}}{D}_{\lambda }^{N-1}\left({\lambda }^{N-1+mn+j}\right)\hfill \\ & =& m^{-s}{\displaystyle \sum _{j=0}^{m-1}}{\displaystyle \sum _{n=0}^{\infty }}{(x+j/m+n)}^{-s}\left({\displaystyle \genfrac{}{}{0pt}{}{N-1+mn+j}{N-1}}\right){\lambda }^{mn+j}\hfill \\ & =& m^{-s}{\displaystyle \sum _{j=0}^{m-1}}{\displaystyle \sum _{n=0}^{\infty }}{(x+j/m+n)}^{-s}\left(\sum _{ml+k=mn+j}\left({\displaystyle \genfrac{}{}{0pt}{}{N-1+l}{N-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right)\right){\lambda }^{mn+j}\hfill \\ & =& m^{-s}{\displaystyle \sum _{k,l=0}^{\infty }}{(x+l+k/m)}^{-s}\left({\displaystyle \genfrac{}{}{0pt}{}{N-1+l}{N-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^{ml+k}.\hfill \end{array}$$

Thanks to Lemma 1 , we obtain

$$\begin{array}{ccc}\hfill {\zeta }_N(s,mx,\lambda )& =& m^{-s}{\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{\displaystyle \sum _{l=0}^{\infty }}{(x+l+k/m)}^{-s}\left({\displaystyle \genfrac{}{}{0pt}{}{N-1+l}{N-1}}\right){\lambda }^{ml}\hfill \\ & =& m^{-s}{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{\zeta }_N(s,x+k/m,{\lambda }^m).\hfill \end{array}$$

This completes the proof. □

Observe that the above theorem has been proved for $N=1$ by Srivastava in ([19] (3.8)) and ([20] p.339, (15)).

Theorem 3

(Inversion formula). Let x be a positive real, m be a positive integer and set ${\displaystyle {\omega }_m=e^{2i\pi /m}}$. Then for any nonnegative integer $j\le m-1$, we have

$$\begin{array}{ccc}\hfill & {\displaystyle m^{s-1}{\displaystyle \sum _{n=0}^{m-1}}{\omega }_m^{-jn}{\zeta }_N(s,x,{\omega }_m^n\lambda )=}& \\ & \\ & {\displaystyle {\displaystyle \sum _{k=0}^{⌊\frac{N(m-1)-j}{m}⌋}}\left(\genfrac{}{}{0pt}{}{N,m}{km+j}\right){\lambda }^{km+j}{\zeta }_N(s,\frac{x+j}{m}+k,{\lambda }^m).}& \end{array}$$

We mention that when $N=1$ and $\left|\lambda \right|=1$, this inversion formula (13) was studied by Chan-Ha in ([8] Theorem 1), and Nakamura ([15] p.288, Theorem 2).

First proof.

Using the multiplication formula again:

$${\zeta }_N(s,mx,\lambda )=m^{-s}{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{\zeta }_N\left(s,x+{\displaystyle \frac{k}{m}},{\lambda }^m\right),$$

(13)

where

$$S_j:={\displaystyle \sum _{n=0}^{m-1}}{\omega }_m^{-jn}{\zeta }_N(s,x,{\omega }_m^n\lambda ),$$

(14)

after some calculation

$$S_j=m^{-s}{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{\zeta }_N\left(s,x+{\displaystyle \frac{k}{m}},{\lambda }^m\right){\displaystyle \sum _{n=0}^{m-1}}{\omega }_m^{n(k-j)}.$$

(15)

Using the following equation

$${\displaystyle \sum _{n=0}^{m-1}}{\omega }_m^{n(k-j)}=\left\{\begin{array}{cc}m,\hfill & \mathrm{if}k\equiv jmodm\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

So only $k=km+j$ remains:

$$S_j=m^{1-s}{\lambda }^{-j}{\displaystyle \sum _{k=0}^{⌊{\displaystyle \frac{N(m-1)-j}{m}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{km+j}}\right){\lambda }^{km}{\zeta }_N\left(s,x+{\displaystyle \frac{j}{m}}+k,{\lambda }^m\right).$$

(16)

□

Second proof.

By the multiplication formula (12), for any non-negative integer $j<m$, we have

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{m-1}}{\omega }^{-jn}{\zeta }_N(s,x,{\omega }^n\lambda )& =& m^{-s}{\displaystyle \sum _{n=0}^{m-1}}{\omega }^{-jn}{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{\zeta }_N(s,x+{\displaystyle \frac{k}{m}},{\lambda }^m)\hfill \\ & =& m^{-s}{\displaystyle \sum _{n=0}^{m-1}}{\omega }^{-jn}{\displaystyle \sum _{l=0}^{m-1}}{\omega }^{nl}{\lambda }^l\left({\displaystyle \sum _{k=0}^{\left[\right(N(m-1)-l)/m]}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{km+l}}\right){\lambda }^{km}{\zeta }_N(s,k+{\displaystyle \frac{x+l}{m}},{\lambda }^m)\right).\hfill \end{array}$$

We then invert the sums and get

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{m-1}}{\omega }^{-jn}{\zeta }_N(s,x,{\omega }^n\lambda )& =& m^{-s}{\displaystyle \sum _{l=0}^{m-1}}\left({\displaystyle \sum _{n=0}^{m-1}}{\omega }^{n(l-j)}\right){\lambda }^l\left({\displaystyle \sum _{k=0}^{\left[\right(N(m-1)-l)/m]}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{km+l}}\right){\lambda }^{km}{\zeta }_N(s,k+{\displaystyle \frac{x+l}{m}},{\lambda }^m)\right).\hfill \end{array}$$

But ${\displaystyle {\displaystyle \sum _{n=0}^{m-1}}{\omega }^{n(l-j)}=m{\delta }_{j,l}}$, so

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{m-1}}{\omega }^{-jn}{\zeta }_N(s,x,{\omega }^n\lambda )& =& m^{-s+1}{\lambda }^j{\displaystyle \sum _{k=0}^{\left[\right(N(m-1)-l)/m]}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{km+j}}\right){\lambda }^{km}{\zeta }_N(s,k+{\displaystyle \frac{x+j}{m}},{\lambda }^m)\hfill \end{array}$$

and the proof is complete. □

For $N=1$, the formula (13) reduces to

$$\begin{array}{c}\hfill \Phi ({\lambda }^m,s,(x+j)/m)=m^{s-1}{\lambda }^{-j}{\displaystyle \sum _{n=0}^{m-1}}{\omega }_m^{-jn}\Phi ({\omega }_m^n\lambda ,s,x).\end{array}$$

(17)

We deduce, from Theorem 2 the following duplication formula.

Corollary 1

(Duplication formulas). Let N be a positive integer. For $m=2$, we obtain

$$\begin{array}{c}\hfill {\zeta }_N(s,x,\lambda )=2^{-s}{\displaystyle \sum _{k=0}^N}\left({\displaystyle \genfrac{}{}{0pt}{}{N}{k}}\right){\lambda }^k{\zeta }_N(s,{\displaystyle \frac{x+k}{2}},{\lambda }^2)\end{array}$$

and $N=1$ gives the well-known identity

$$\begin{array}{c}\hfill 2^s\Phi (\lambda ,s,x)=\Phi ({\lambda }^2,s,x/2)+\lambda \Phi ({\lambda }^2,s,(x+1)/2).\end{array}$$

3. New Formulas on Higher Apostol-Bernoulli and Apostol-Euler Polynomials

The function $s\to {\zeta }_N(s;x,\lambda )$ has analytic continuation to whole complex numbers, except eventual poles at $s=1,\cdots ,N.$ At $s=-n<0$ negative integers, we have the relations.

$$\begin{array}{c}\hfill E_n^{\left(N\right)}(x;\lambda )=2^N{\zeta }_N(-n,x,-\lambda )\end{array}$$

and

$$\begin{array}{c}\hfill B_{n+N}^{\left(N\right)}(x;\lambda )={(-1)}^N<n+1{>}_N{\zeta }_N(-n,x,\lambda )\end{array}$$

see for details [21]. By using the above formulas, we give the following results.

Theorem 4

(generalized Multiplication formulae). Let $n,m$ be positive integers, then we have

$$\begin{array}{ccc}\hfill B_{n+N}^{\left(N\right)}(mx;\lambda )& =& m^n{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{B}_{n+N}^{\left(N\right)}(x+{\displaystyle \frac{k}{m}};{\lambda }^m)\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill E_n^{\left(N\right)}(mx;\lambda )& =& m^n{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){(-1)}^k{\lambda }^k{E}_n^{\left(N\right)}(x+{\displaystyle \frac{k}{m}};{\lambda }^m),\mathit{if}m\mathit{odd}\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill E_n^{\left(N\right)}(mx;\lambda )& =& {\displaystyle \frac{{(-2)}^N{m}^n}{<n+1{>}_N}}{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){(-1)}^k{\lambda }^k{B}_{n+N}^{\left(N\right)}(x+{\displaystyle \frac{k}{m}};{\lambda }^m),\mathit{if}m\mathit{even}.\hfill \end{array}$$

(20)

This result improves a similar result found by Carlitz in [7].

Proof. 

Let $n,m$ be positive integers, then we have

$$\begin{array}{ccc}\hfill B_{n+N}^{\left(N\right)}(mx;\lambda )& =& {(-1)}^N<n+1{>}_N{\zeta }_N(-n,mx,\lambda )\hfill \\ & =& m^n{(-1)}^N<n+1{>}_N{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){(-\lambda )}^k{\zeta }_N(-n,x+{\displaystyle \frac{k}{m}},{\lambda }^m)\hfill \\ & =& m^n{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{B}_{n+N}^{\left(N\right)}(x+{\displaystyle \frac{k}{m}};{\lambda }^m)\hfill \end{array}$$

and also we have

$$\begin{array}{ccc}\hfill E_n^{\left(N\right)}(mx;\lambda )& =& 2^N{\zeta }_N(-n,mx,-\lambda )\hfill \\ & =& m^n{2}^N{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){(-\lambda )}^k{\zeta }_N(-n,x+{\displaystyle \frac{k}{m}},{(-\lambda )}^m)\hfill \\ & =& m^n{2}^N{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){(-1)}^k{\lambda }^k{\zeta }_N(-n,x+{\displaystyle \frac{k}{m}},{(-\lambda )}^m).\hfill \end{array}$$

Now, if m is odd, we have

$$\begin{array}{c}\hfill {\zeta }_N(-n,x+{\displaystyle \frac{k}{m}},{(-\lambda )}^m)={\zeta }_N(-n,x+{\displaystyle \frac{k}{m}},-{\lambda }^m)=2^{-N}{E}_n^{\left(N\right)}(x+{\displaystyle \frac{k}{m}};{\lambda }^m)\end{array}$$

and, if m is even, we have

$$\begin{array}{c}\hfill {\zeta }_N(-n,x+{\displaystyle \frac{k}{m}},{(-\lambda )}^m)={\zeta }_N(-n,x+{\displaystyle \frac{k}{m}},{\lambda }^m)={\displaystyle \frac{{(-1)}^N}{<n+1{>}_N}}{B}_{n+N}^{\left(N\right)}(x+{\displaystyle \frac{k}{m}};{\lambda }^m).\end{array}$$

Thus the proof is complete. □

Theorem 5

(Generalized difference formulae). Let $n,m$ be positive integers, then we have

$$\begin{array}{c}\hfill {\displaystyle \sum _{j=0}^N}\left({\displaystyle \genfrac{}{}{0pt}{}{N}{j}}\right){(-1)}^{(m+1)j}{\lambda }^{mj}{E}_n^{\left(N\right)}(x+mj;\lambda )=2^{-N}{\displaystyle \sum _{k=0}^{N(m-1)}}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{(x+k)}^n\\ \hfill {\displaystyle \sum _{j=0}^N}\left({\displaystyle \genfrac{}{}{0pt}{}{N}{j}}\right){(-1)}^j{\lambda }^{mj}{B}_{n+N}^{\left(N\right)}(x+mj;\lambda )={\displaystyle \frac{{(-1)}^N}{<n+1{>}_n}}{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{(x+k)}^n.\end{array}$$

Proof. 

Start from the known definitions:

$$E_n^{\left(N\right)}(x;\lambda )=2^N{\zeta }_N(-n,x,-\lambda ),B_{n+N}^{\left(N\right)}(x;\lambda )={(-1)}^N{\langle n+1\rangle }_N{\zeta }_N(-n,x,\lambda ).$$

Apply the difference operator to the left-hand side:

$${\Delta }_m^{N}f\left(x\right):={\displaystyle \sum _{j=0}^N}\left({\displaystyle \genfrac{}{}{0pt}{}{N}{j}}\right){(-1)}^{j}f(x+mj).$$

Now consider the sum:

$${\displaystyle \sum _{j=0}^N}\left({\displaystyle \genfrac{}{}{0pt}{}{N}{j}}\right){(-1)}^{(m+1)j}{\lambda }^{mj}{E}_n^{\left(N\right)}(x+mj;\lambda ).$$

Substitute using the zeta relation:

$$=2^N{\displaystyle \sum _{j=0}^N}\left({\displaystyle \genfrac{}{}{0pt}{}{N}{j}}\right){(-1)}^{(m+1)j}{\lambda }^{mj}{\zeta }_N(-n,x+mj,-\lambda ).$$

Use the generating function expansion for ${\zeta }_N(-n,mx,-\lambda )$, as in Theorem 2.1:

$${\zeta }_N(-n,mx,-\lambda )=m^n{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){(-1)}^k{\lambda }^k{(x+k)}^n.$$

Divide both sides by $2^N$, and you get the desired expression for Euler.

Similarly, for the Bernoulli case:

$${\displaystyle \sum _{j=0}^N}\left({\displaystyle \genfrac{}{}{0pt}{}{N}{j}}\right){(-1)}^j{\lambda }^{mj}{B}_{n+N}^{\left(N\right)}(x+mj;\lambda )={(-1)}^N{\langle n+1\rangle }_N{\displaystyle \sum _{k=0}^{N(m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k}}\right){\lambda }^k{(x+k)}^n.$$

□

4. New Identities on the Polynomial Coefficients

In this section, we investigate some properties of symmetry, difference and recursion formulas, unimodality, and asymptotic behavior for the polynomial coefficients. We state the main results on the polynomial coefficients.

4.1. Symmetry, Difference and Recursion Formulas

Theorem 6

(Symmetry, Difference, Recursion). We have the symmetry formula

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{N(q-1)-k}}\right)\mathrm{for}0\le k\le N(q-1).\end{array}$$

(21)

For any non-negative integer $k\le (N+1)(m-1)$, we have the recurrence formula

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{N+1,m}{k}}\right)={\displaystyle \sum _{j=0}^{m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k-j}}\right).\end{array}$$

(22)

Let k be a positive integer such that $k\le N(q-1)$, then we have the difference formula

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{N+1,q}{k}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{N+1,q}{k-1}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k-q}}\right).\end{array}$$

(23)

For any non-negative integer $k\le (N+1)(q-1)-1$, we have another recursion formula

$$\begin{array}{c}\hfill (k+1)\left({\displaystyle \genfrac{}{}{0pt}{}{N+1,q}{k+1}}\right)=(N+1){\displaystyle \sum _{j=0}^{q-2}}(j+1)\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k-j}}\right).\end{array}$$

(24)

Proof. 

From the expansions in the the equality

$$\begin{array}{c}\hfill t^{N(q-1)}{\left({\displaystyle \frac{1-{(1/t)}^q}{1-(1/t)}}\right)}^N={\left({\displaystyle \frac{1-t^q}{1-t}}\right)}^N,\end{array}$$

we deduce the symmetry formula

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{N(q-1)-k}}\right)\mathrm{for}0\le k\le N(q-1).\end{array}$$

(25)

For any non-negative integer $k\le (N+1)(m-1)$, we have

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{N+1,m}{k}}\right)={\displaystyle \sum _{j=0}^{m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k-j}}\right).\end{array}$$

(26)

Indeed, we just say that ${(1+t+\dots +t^{m-1})}^{N+1}=(1+t+...+t^{m-1}){(1+t+\dots +t^{m-1})}^N$

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{N+1,m}{k}}\right)={\displaystyle \sum _{j=0}^{m-1)}}\left({\displaystyle \genfrac{}{}{0pt}{}{N,m}{k-j}}\right).\end{array}$$

(27)

Indeed, we just say that ${(1+t+\dots +t^{m-1})}^{N+1}=(1+t+...+t^{m-1}){(1+t+\dots +t^{m-1})}^N$.

We then obtain general recursion formula.

Let k be a positive integer such that $k\le N(q-1)$, then we have

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{N+1,q}{k}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{N+1,q}{k-1}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k-q}}\right),\end{array}$$

(28)

where the $\delta$ is the Kronecker’s symbol. Here, the expansions considered are those of the equality

$$\begin{array}{c}\hfill (1-t){\left({\displaystyle \frac{1-t^q}{1-t}}\right)}^{N+1}=(1-t^q){\left({\displaystyle \frac{1-t^q}{1-t}}\right)}^N.\end{array}$$

Thus, we have our difference formula.

For any non-negative integer $k\le (N+1)(q-1)-1$, from the expansions in the derivative equality

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{(1+t+\dots +t^{q-1})}^{N+1}=(N+1)(1+2t+..+(q-1)t^{q-2}){(1+t+..+t^{q-1})}^N,\end{array}$$

we obtain

$$\begin{array}{c}\hfill (k+1)\left({\displaystyle \genfrac{}{}{0pt}{}{N+1,q}{k+1}}\right)=(N+1){\displaystyle \sum _{j=0}^{q-2}}(j+1)\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k-j}}\right).\end{array}$$

(29)

Then we get the second recurrence formula. □

4.2. Unimodality

The polynomial coefficients satisfy the unimodality chain.

Theorem 7

(Unimodality). Let N and q be any positive integers, the polynomial coefficients satisfy the unimodality condition

$$\begin{array}{c}\hfill 1\le \left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{1}}\right)\le \cdots \le \left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k_{max}}}\right)\end{array}$$

(30)

and

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k_{max}}}\right)\ge \left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k_{max}+1}}\right)\ge \cdots \ge \left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{N(q-1)-1}}\right)\ge 1,\end{array}$$

(31)

where

$$k_{max}=\left[{\displaystyle \frac{N(q-1)}{2}}\right].$$

Furthermore, when $N(q-1)$ is odd, we have the equality $\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k_{max}}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k_{max}+1}}\right)$ (representing a plateau of two points), if $N(q-1)$ is even $k_{max}$ is indicating a peak.

Proof. 

Several methods can be used to prove unimodality. The unimodality result can be derived from Andrew’s work in [1, Theorem 3.9, p.45-46]. Indeed, the polynomial $p\left(t\right)=1+t+\dots +t^{q-1}$ is, for any positive integer q, clearly a unimodal and reciprocal polynomial with non-negative coefficients. Hence, repeated application of the Andrew’s theorem shows that the polynomials ${\left(p\left(t\right)\right)}^2,\cdots ,{\left(p\left(t\right)\right)}^N,\dots$ are of the same type. □

Theorem 8

(Log-concavity inequalities). The coefficients of the polynomial $p\left(t\right)$ are all positive and satisfy the log-concavity inequalities. Consequently, the coefficients of ${\left(p\left(t\right)\right)}^2,\cdots ,{\left(p\left(t\right)\right)}^N,\cdots$ also exhibit the same properties. In other words, we have:

$$\begin{array}{c}\hfill {\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)}^2\ge \left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k+1}}\right)\mathit{for}1\le k\le N(q-1)-1.\end{array}$$

(32)

Proof. 

The proof of this result can be derived from Stanley’s work [22, Proposition 2]. The same arguments also applies to the log-concavity property. For more details see [22, Proposition 2] for further details. Since the coefficients of the polynomial $p\left(t\right)$ are all postive and satisfy the log-concavity inequalities, the same holds for the coefficients of ${\left(p\left(t\right)\right)}^2,\cdots ,{\left(p\left(t\right)\right)}^N,\cdots$. In particular, we have

$$\begin{array}{c}\hfill {\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)}^2\ge \left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k+1}}\right)\mathit{for}1\le k\le N(q-1)-1.\end{array}$$

(33)

□

4.3. Asymptotic Behavior for the Polynomial Coefficients

In this section, we derive an asymptotic formula for the polynomial coefficients. To this end, it is convenient to recall a generalization of Sperner’s formula for polynomial coefficients due to Sander [17].

Theorem 9.

For any non-negative integer $k\le N(q-1)$, we have the integral representation

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)={\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}{\left({\displaystyle \frac{sinqu}{sinu}}\right)}^{N}cos(N(q-1)-2k)udu.\end{array}$$

(34)

Moreover, the greatest coefficient is given by

$$\begin{array}{c}\hfill {\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)}_{max}=\left\{\begin{array}{cc}{\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}{\left({\displaystyle \frac{sinqu}{sinu}}\right)}^{N}du& \mathit{if}N(q-1)\mathit{is} \mathit{even},\\ {\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2}{\left({\displaystyle \frac{sinqu}{sinu}}\right)}^{N}cosudu& \mathit{if}N(q-1)\mathit{is} \mathit{odd}.\end{array}\right.\end{array}$$

(35)

It is easy to see that the maximal coefficient is attained at

$$\begin{array}{c}\hfill k_{max}=\left[{\displaystyle \frac{N(q-1)}{2}}\right].\end{array}$$

(36)

When $q=2$ we recover Sperner’s theorem, see [18]. Formula (34) was proved by Sander in [17].

As $N\to +\infty$ with $q>1$ fixed, the maximal coefficient has the asymptotic behavior

$$\begin{array}{c}\hfill {\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)}_{max}\sim q^N\sqrt{{\displaystyle \frac{6}{(q^2-1)\pi N}}}.\end{array}$$

(37)

This result is stated in [10, p.78] without proof. A natural question arises: what is the asymptotic behavior of $\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)$ for an arbitrary integer, not only for $k=k_{max}$?

In the following sections, we address this question. As a corollary, the case $k=k_{max}$, yields a proof of (37).

Theorem 10.

Fix the integer $q\ge 2$ and let the integers $0<k<N(q-1)$ increase to infinity, we have

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)\sim q^N\sqrt{{\displaystyle \frac{6}{(q^2-1)\pi N}}}exp\left(-{\displaystyle \frac{3{(2k-N(q-1))}^2}{2(q^2-1)N}}\right).\end{array}$$

(38)

Proof. 

Using (34), we first write

$$\begin{array}{cc}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)=& {\displaystyle \frac{2}{\pi }}{\int }_0^{\pi /2q}{\left({\displaystyle \frac{sinqu}{sinu}}\right)}^{N}cos(N(q-1)-2k)udu\\ & +{\displaystyle \frac{2}{\pi }}{\int }_{\pi /2q}^{\pi /2}{\left({\displaystyle \frac{sinqu}{sinu}}\right)}^{N}cos(N(q-1)-2k)udu.\end{array}$$

On one hand, as the function $u\mapsto 1/sinu$ is decreasing on $\left[{\displaystyle \frac{\pi }{2q}},{\displaystyle \frac{\pi }{2}}\right]$, we get the estimates

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{2}{\pi }}{\int }_{\pi /2q}^{\pi }{\left({\displaystyle \frac{sinqu}{sinu}}\right)}^{N}cos(N(q-1)-2k)udu\right|\le {\displaystyle \frac{2}{\pi {sin}^N(\pi /2q)}}{\int }_{\pi /2q}^{\pi /2}\mid sinqu{\mid }^{N}du.\end{array}$$

(39)

Then

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{2}{\pi }}{\int }_{\pi /2q}^{\pi }{\left({\displaystyle \frac{sinqu}{sinu}}\right)}^{N}cos(N(q-1)-2k)udu\right|\le {\displaystyle \frac{(q-1)}{q{sin}^N(\pi /2q)}}.\end{array}$$

(40)

Hence, for the asymptotic, the important contribution in the integral representing $\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)$ is in the near of 0.

The function $u\mapsto sin\left(qu\right)/sinu$ is positive on $[0,\pi /2q]$, so we can write

$$\begin{array}{c}\hfill q^{-N}{\int }_0^{\pi /2q}{\left({\displaystyle \frac{sin\left(qu\right)}{sinu}}\right)}^{N}cos(N(q-1)-2k)udu={\int }_0^{\pi /2q}{e}^{-Nh\left(u\right)}cos(n(q-1)-2k)udu\end{array}$$

with

$$\begin{array}{c}\hfill h\left(u\right)=log\left({\displaystyle \frac{qsinu}{sinqu}}\right).\end{array}$$

To expand this function as a serie, we have

$$\begin{array}{cc}\hfill log\left({\displaystyle \frac{sinz}{z}}\right)& =log{\displaystyle \prod _{n=1}^{\infty }}\left(1-{\displaystyle \frac{z^2}{n^2{\pi }^2}}\right)={\displaystyle \sum _{n=1}^{\infty }}log\left(1-{\displaystyle \frac{z^2}{n^2{\pi }^2}}\right)\hfill \\ & =-{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{1}{k}}{\left({\displaystyle \frac{z}{n\pi }}\right)}^{2k}=-{\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1}{k}}{\left({\displaystyle \frac{z}{n\pi }}\right)}^{2k}\hfill \\ & =-{\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{\zeta \left(2k\right)}{k{\pi }^{2k}}}{z}^{2k}=-{\displaystyle \frac{1}{6}}{z}^2-{\displaystyle \frac{1}{180}}{z}^4-{\displaystyle \frac{1}{2835}}{z}^6-{\displaystyle \frac{1}{37800}}{z}^8-\cdots \hfill \end{array}$$

this power series converges when $\left|z\right|<\pi$. Note that

$$\zeta \left(2k\right)={\displaystyle \frac{{(-1)}^{k+1}{B}_{2k}{\left(2\pi \right)}^{2k}}{2\left(2k\right)!}}.$$

Therefore, we get

$$\begin{array}{c}\hfill log\left({\displaystyle \frac{sinz}{z}}\right)=-{\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{2^{2k-1}\left|B_{2k}\right|}{k\left(2k\right)!}}{z}^{2k}\mathrm{for}0<\left|z\right|<\pi .\end{array}$$

Thus for all $u\in ]0,\pi /2q[$, $h\left(u\right)$ has the expansion

$$\begin{array}{c}\hfill h\left(u\right)={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{2^{2k-1}\left|B_{2k}\right|}{k\left(2k\right)!}}(q^{2k}-1)u^{2k}:=a_q{u}^2+u^4{g}_q\left(u\right),\end{array}$$

where $a_q=(q^2-1)/6$ and $g_q$ a remaining function. Let us write

$$\begin{array}{ccc}& {\int }_0^{\pi /2q}{e}^{-Nh\left(u\right)}cos(N(q-1)-2k)udu=& \\ & {\int }_0^{\pi /2q}{e}^{-N{a}_q{u}^2}cos(N(q-1)-2k)udu& \\ & -{\int }_0^{\pi /2q}{e}^{-N{a}_q{u}^2}\left(1-e^{-N{u}^4{g}_q\left(u\right)}\right)cos(N(q-1)-2k)udu.\end{array}$$

Now, if ${\displaystyle M_q=\underset{[0,\pi /2q]}{max}\mid g_q\left(u\right)\mid }$, we have

$$\begin{array}{ccc}& \left|{\int }_0^{\pi /2q}{e}^{-N{a}_q{u}^2}\left(1-e^{-N{u}^4{g}_q\left(u\right)}\right)cos(N(q-1)-2k)udu\right|& \\ & \le {\int }_0^{\pi /2q}{e}^{-N{a}_q{u}^2}(1-e^{-N{u}^4{g}_q\left(u\right)})du\end{array}$$

and

$$\begin{array}{ccc}& {\int }_0^{\pi /2q}{e}^{-N{a}_q{u}^2}(1-e^{-N{u}^4{g}_q\left(u\right)})du\le N{M}_q{\int }_0^{\pi /2q}{u}^4{e}^{-N{a}_q{u}^2}du,& \end{array}$$

$$\begin{array}{c}\hfill {\int }_0^{\pi /2q}{u}^4{e}^{-N{a}_q{u}^2}du\le {\displaystyle \frac{3}{4N^2{a}_q^2}}{\int }_0^{\infty }{e}^{-N{a}_q{u}^2}du={\displaystyle \frac{3}{8N^2{a}_q^2}}\sqrt{{\displaystyle \frac{\pi }{N{a}_q}}}={\displaystyle \frac{3\sqrt{\pi }}{8N^{5/2}{a}_q^{3/2}}}.\end{array}$$

These inequalities gives

$$\begin{array}{ccc}\hfill & \left|{\int }_0^{\pi /2q}{e}^{-N{a}_q{u}^2}\left(1-e^{-N{u}^4{g}_q\left(u\right)}\right)cos(N(q-1)-2k)udu\right|& \\ & \le {\displaystyle \frac{3M_q\sqrt{\pi }}{8N^{3/2}{a}_q^{3/2}}}=O\left(N^{-3/2}\right).& \hfill \end{array}$$

(41)

We thus, from the above equations (40) and (41), as $N\to \infty$, we get our desired estimate

$$\left({\displaystyle \genfrac{}{}{0pt}{}{N,q}{k}}\right)\sim q^N\sqrt{{\displaystyle \frac{6}{(q^2-1)\pi N}}}exp\left(-{\displaystyle \frac{3{(2k-N(q-1))}^2}{2(q^2-1)N}}\right).$$

□

5. Conclusion

In this work, we have established several fundamental properties of the N-tuple Hurwitz-Lerch zeta functions ${\zeta }_N(s,x,\lambda )$ for any positive integer N and complex parameter $\lambda$. In particular, we derived multiplication and inversion formulas involving the associated polynomial coefficients. Beyond these results, we obtained new identities and relations for higher–order Hurwitz–Lerch zeta functions, as well as for higher Apostol–Bernoulli and Apostol–Euler polynomials. Finally, we studied the polynomial coefficients appearing in these formulas. Their symmetry, difference, and recursion properties were established, together with their asymptotic behavior.